The Mixing method: coordinate descent for low-rank semidefinite programming
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In this paper, we propose a coordinate descent approach to low-rank structured semidefinite programming. The approach, which we call the Mixing method, is extremely simple to implement, has no free parameters, and typically attains an order of magnitude or better improvement in optimization performance over the current state of the art. We show that for certain problems, the method is strictly decreasing and guaranteed to converge to a critical point. We then apply the algorithm to three separate domains: solving the maximum cut semidefinite relaxation, solving a (novel) maximum satisfiability relaxation, and solving the GloVe word embedding optimization problem. In all settings, we demonstrate improvement over the existing state of the art along various dimensions. In total, this work substantially expands the scope and scale of problems that can be solved using semidefinite programming methods.
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